Factorial Analysis of Variance

More than 2 Independent More than 2 Independent VariablesVariables

Between-Subjects DesignsBetween-Subjects Designs

 

What is a FactorialWhat is a Factorial

• At least two independent variablesAt least two independent variables

• All combinations of each variable All combinations of each variable (completely crossed)(completely crossed)

• R X C factorialR X C factorial

• CellsCells

Variables and CellsVariables and Cells

• 2 X 22 X 2 2 IVs 4 cells 2 IVs 4 cells

• 2 X 32 X 3 2 IVs 6 cells 2 IVs 6 cells

• 2 X 3 X 32 X 3 X 3 3 IVs 18 cells 3 IVs 18 cells

• 2 X 2 X 2 X 2 4 Ivs 16 cells2 X 2 X 2 X 2 4 Ivs 16 cells

Effects TestedEffects Tested

• 2 X 22 X 2 A, B, A X BA, B, A X B

• 2 X 2 X 22 X 2 X 2 A, B, C, A X B, A, B, C, A X B,

• A X C, B X C, A X B X CA X C, B X C, A X B X C

• 2 X 2 X 2 X 2 A, B, C, D, A X B, A X C, 2 X 2 X 2 X 2 A, B, C, D, A X B, A X C, A X D, B X C, B X D, C X D, A X B X C, A X D, B X C, B X D, C X D, A X B X C, A X B X D, B X C X D, A X B X C X DA X B X D, B X C X D, A X B X C X D

Video ViolenceVideo Violence

• Bushman studyBushman study Two independent variablesTwo independent variables

• Two kinds of videosTwo kinds of videos

• Male and female subjectsMale and female subjects

• See following diagramSee following diagram

2 X 2 Factorial2 X 2 Factorial

ViolentVideo

NonviolentVideo

Male

Female

Bushman’s Study-cont.Bushman’s Study-cont.

• Dependent variable = number of Dependent variable = number of aggessive associatesaggessive associates

• 50 observations in each cell50 observations in each cell

• We will work with means and st. We will work with means and st. dev., instead of raw data.dev., instead of raw data. This illustrates important concepts.This illustrates important concepts.

The Data The Data (cell means and standard (cell means and standard

deviations)deviations)ViolentVideo

NonviolentVideo Means

Male 7.7(4.6)

6.2(3.5)

6.95

Female 6.5(4.2)

5.1(2.8)

5.80

Means 7.1 5.65 6.375

Plotting ResultsPlotting Results

0

2

4

6

8

10

Violent Video Nonviolent Video

Aggre

ssiv

e A

ssoci

ate

s

Male Female

Effects to be estimatedEffects to be estimated• Differences due to videosDifferences due to videos

Violent appear greater than nonviolentViolent appear greater than nonviolent

• Differences due to genderDifferences due to gender Males appear higher than femalesMales appear higher than females

• Interaction of video and genderInteraction of video and gender What is an interaction?What is an interaction?

Does violence affect males and females equally?Does violence affect males and females equally?

Cont.

Estimated Effects--cont.Estimated Effects--cont.

• ErrorError average within-cell varianceaverage within-cell variance

• Sum of squares and mean squaresSum of squares and mean squares Extension of the same concepts in the Extension of the same concepts in the

one-wayone-way

CalculationsCalculations

• Total sum of squaresTotal sum of squares

• Main effect sum of squaresMain effect sum of squares

2..XXSStotal

2..XXngSS Vvideo

2..XXnvSS Ggender

Cont.

Calculations--cont.Calculations--cont.

• Interaction sum of squaresInteraction sum of squares Calculate SSCalculate SScellscells and subtract SS and subtract SSVV and SS and SSGG

• SSSSerrorerror = SS = SStotaltotal - SS - SScellscells

or, or, MSMSerrorerror can be found as average of cell variances can be found as average of cell variances

2..)( XXnSS ijcells

Degrees of FreedomDegrees of Freedom

• dfdf for main effects = number of for main effects = number of levels - 1levels - 1

• dfdf for interaction = product of for interaction = product of dfdfmain main

effectseffects

• dfdf errorerror = = NN - - abab = = NN - # cells - # cells

• dfdftotaltotal = = NN - 1 - 1

Calculations for Bushman Calculations for Bushman DataData

• SSSStotaltotal requires raw data. requires raw data.

It is actually = 171.50It is actually = 171.50

• SSSSvideovideo

125.105

375.665.5375.61.7250

..22

2

XXngSS Vvideo

Cont.

Calculations--cont.Calculations--cont.

• SSSSgendergender

125.66

375.680.5375.695.6)2(50

..22

2

XXnvSS Ggender

Cont.

Calculations--cont.Calculations--cont.

• SSSScellscells

• SSSSVXGVXG = SS = SScellscells - SS - SSvideovideo - SS - SSgendergender

== 171.375 - 105.125 - 66.125 = 0.125 171.375 - 105.125 - 66.125 = 0.125

375.171)4275.3(50

)375.61.5()375.65.6(

)375.62.6()375.67.7(50

..)(

22

22

2

XXnSS cellcells

Cont.

Calculations--cont.Calculations--cont.

• MSMSerrorerror = average of cell variances = = average of cell variances =(4.6(4.622 + 3.5 + 3.522 + 4.2 + 4.222 + 2.8 + 2.822)/4 )/4 =58.89/4 = 14.723 =58.89/4 = 14.723

• Note that this is MSNote that this is MSerrorerror and not SS and not SSerrorerror

Summary TableSummary Table

Source df SS MS FVideo 1 105.125 105.125 7.14Gender 1 66.125 66.125 4.49VXG 1 0.125 0.125 .01Error 196 2885.610 14.723Total 199 3056.980

ConclusionsConclusions

• Main effectsMain effects Significant difference due to videoSignificant difference due to video

• More aggressive associates following More aggressive associates following violent videoviolent video

Significant difference due to genderSignificant difference due to gender• Males have more aggressive associates Males have more aggressive associates

than females.than females.

Cont.

Conclusions--cont.Conclusions--cont.

• InteractionInteraction No interaction between video and No interaction between video and

gendergender• Difference between violent and Difference between violent and

nonviolent video is the same for males nonviolent video is the same for males (1.5) as it is for females (1.4)(1.5) as it is for females (1.4)

• We could see this in the graph of the We could see this in the graph of the data.data.

Elaborate on InteractionsElaborate on Interactions

• Diagrammed on next slide as line graphDiagrammed on next slide as line graph

• Note parallelism of linesNote parallelism of lines Means video differences did not depend on Means video differences did not depend on

gendergender

• A significant interaction would have A significant interaction would have nonparallel linesnonparallel lines Ordinal and disordinal interactionsOrdinal and disordinal interactions

Line Graph of InteractionLine Graph of Interaction

0123456789

Violent Video Nonviolent Video

Aggre

ssiv

e A

ssoci

ate

s

MaleFemale

Simple EffectsSimple Effects

• Effect of one independent variable Effect of one independent variable at one level of the other.at one level of the other.

• e.g. Difference between males and e.g. Difference between males and females for only violent videofemales for only violent video

• Difference between males and Difference between males and females for only nonviolent videofemales for only nonviolent video

Unequal Sample SizesUnequal Sample Sizes

• A serious problem for hand A serious problem for hand calculationscalculations

• Can be computed easily using Can be computed easily using computer softwarecomputer software

• Can make the interpretation difficultCan make the interpretation difficult Depends, in part, on why the data are Depends, in part, on why the data are

missing.missing.

Magnitude of EffectMagnitude of Effect

• Eta SquaredEta Squared

InterpretationInterpretation

• Omega squaredOmega squared Less biased estimateLess biased estimate

total

effect

SS

SS2

errortotal

erroreffect

MSSS

MSkSS

)1(2

k = number of levels for the effectin question

Cont.

Effect Size—cont.Effect Size—cont.

• As with one-way, we can calculate As with one-way, we can calculate effect size for each kind of effect effect size for each kind of effect separately.separately.

• Most sensible to stick to Most sensible to stick to comparisons of two groups.comparisons of two groups.

• Same formulae as for Same formulae as for tt tests. tests.